All Triples Independent but Quadruple Not

Let $\Omega = \{0, 1, 2, \ldots, 7\}$ with uniform probability $P(\{\omega\}) = 1/8$. Write each $\omega$ in binary as $(b_2, b_1, b_0)$. Define events: $$A = \{\omega : b_0 = 1\}, \quad B = \{\omega : b_1 = 1\}, \quad C = \{\omega : b_2 = 1\}, \quad D = \{\omega : b_0 \oplus b_1 \oplus b_2 = 1\}.$$ (a) Show that $A$, $B$, $C$ are mutually independent. (b) Show that any triple chosen from $\{A,B,C,D\}$ is mutually independent. (c) Show that $A$, $B$, $C$, $D$ are not mutually independent.